suppose I have the following quadratic form $$ {\bf{x^{\top}Ax}} = constant $$ where A is positive definite and I know that $$ {{\bf{x}}^ \top }{\bf{1}} = 1 $$
Is there an analytical solution to this problem. I can reduce it to the following form $$ {\bf{y}} = {A^{\frac{1}{2}}}{\bf{x}}{\rm{\space\space\space\space\space st:\space }}{{\bf{y}}^ \top }{\bf{y}} = 1 $$ and further reduce it to $$ {{\bf{1}}^ \top }{{\bf{A}}^{ - \frac{1}{2}}}{\bf{y}} = 1 $$
but I don't know how to proceed from here. Any suggestions besides using numerical methods ?